We propose a dynamically bi-orthogonal
method (DyBO) to study time dependent stochastic partial differential equations
(SPDEs). The objective of our method is to exploit some intrinsic sparse
structure in the stochastic solution by constructing the sparsest
representation of the stochastic solution via a bi-orthogonal basis. It is
well-known that the Karhunen-Loeve expansion minimizes the total mean squared
error and gives the sparsest representation of stochastic solutions. However,
the computation of the KL expansion could be quite expensive. In this talk, we
derive an equivalent system that governs the evolution of the spatial and
stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs
the reduced basis on-the-fly without the need to form the covariance matrix or
to compute its eigen-decomposition. Several numerical experiments will be
provided to demonstrate the effectiveness of the DyBO method.

Bio:

Zhiwen Zhang is an assistant professor in
the university of Hong Kong. He was a postdoctoral scholar in the Department of
Computing and Mathematical Sciences, California Institute of Technology from
2011 to 2015. He graduated from the Department of Mathematical Sciences,
Tsinghua University in 2011, where he was awarded the degree of Ph.D. in
Applied Mathematics. From 2008 to 2009, he studied in the University of
Wisconsin at Madison as a visiting PhD student. His research interests lie in
the applied analysis and numerical computation of problems arising from quantum
chemistry, wave propagation, porous media, biomechanics, sparse Bayesian
learning, and uncertainty quantification (UQ) for stochastic fluid dynamics and
random heterogeneous media.